On 07/09/2012 06:33 AM, Michael Ossipoff wrote:
SL/Webster minimizes the SL index, right? It's known that Webster has
_no_ bias if the distribution-condition that I described obtains--the
uniform distribution condition.
I'm not a statistician either, and so this is just a tentative
possibility suggestion: What about finding, by trial and error, the
allocation that minimizes the calculated correlation measure. Say, the
Pearson correlation, for example. Find by trial and error the allocation
with the lowest Pearson correlation between q and s/q.
For the goal of getting the best allocation each time (as opposed to
overall time-averaged equality of s/q), might that correlation
optimization be best?
Sure, you could empirically optimize the method. If you want
population-pair monotonicity, then your task becomes much easier: only
divisor methods can have it so you just have to find the right parameter
for the generalized divisor method:
f(x,g) = floor(x + g(x))
where g(x) is within [0...1] for all x, and one then finds a divisor so
that x_1 = voter share for state 1 / divisor, so that sum over all
states is equal to the number of seats.
We may further restrict ourselves to a "somewhat" generalized divisor
method:
f(x, p) = floor(x + p).
For Webster, p = 0.5. Warren said p = 0.495 or so would optimize in the
US (and it might, I haven't read his reasoning in detail). Also, I think
that the bias is monotone with respect to p. At one end you have
f(x) = floor(x + 0) = floor(x)
which is Jefferson's method (D'Hondt) and greatly favors large states.
At the other, you have
f(x) = floor(x + 1) = ceil(x)
which is Adams's method and greatly favors small states.
If f(x, p) is monotone with respect to bias as p is varied, then you
could use any number of root-finding algorithms to find the p that sets
bias to zero, assuming your bias measure is continuous. Even if it's not
continuous, you could find p so that decreasing p just a little leads
your bias measure to report large-state favoritism and increasing p just
a little leads your bias measure to report small-state favoritism.
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